Sorensen’s Proof and my response (new and improved… hopefully)

What happens when we try to apply (C.b*) to Sorensen’s proof? Well, it would seem that the premise is a contradiction: 

1. B_a[à($p)($x)(~àp & B_xp)]               (Sorensen’s Premise)

2. à($p)($x)(~àp & B_xp) Î m              (1, (C.b*))

3. ($p)($x)(~àp & B_xp) Î m*              (2, since ($p)($x)(~àp & B_xp) is possible in m then there must be

                                                            at least one model (world) such that m has access to that model

                                                            and ($p)($x)(~àp & B_xp) is a member of that model)

4. ($x)(~àq & B_xq) Î m*                     (3, Existential Instantiation)

5. ~àq & B_bq Î m*                              (4, Existential Instantiation)

6. ÿ~q Î m*                                         (5, Simplification and Rule of ÿ-à Interchange)

7. B_bq Î m*                                         (5, Simplification)

8. q Î m**                                           (7, (C.b*))

9. ~q Î m**                                         (6, since ~q is necessary in m* then all models (worlds) are

                                                            such that if m* has access to that model then ~q is a

                                                            member of that model)

Now one might not be very convinced with the proof that I have just given. They could say that (C.b*) does not accurately apply the LNC in a belief context or the fact that within my proof that I use doxastic possible worlds and metaphysical possible worlds the same way.  That (C.b*) assumes that Sorensen is wrong to begin with so my proof just begs the question.

            Although I treat doxastic possible worlds and metaphysically possible worlds the same there are going to be some instances where they match up (when the beliefs are true for example). This is certainly true for cases involving the LNC. If the LNC holds then it holds everywhere, doxastic possible worlds and metaphysical possible worlds. For those who do not accept the LNC then the law will not hold equally in both types of worlds. Sorensen wants to have it both ways, he wants to maintain the LNC and maintain that people can and do have contradictory beliefs. To say that the LNC only fails with respect to belief is some how making beliefs unique. To treat beliefs in this manner appears very ad hoc and we shouldn’t let our intuitions about belief be justification for this ad hoc move.

The other problem with my proof has to do with whether employing (C.b*) is begging the question or not. Obviously, there is an attempt within the sciences and the philosophy of mind to try and explain what beliefs are. If we assume that people can and do have contradictory beliefs then, in principle, there is no way to explain what beliefs are that includes the LNC.[1] We will never be able to employ something like (C.b*) within our explanation of beliefs. If that is the case then no consistent explanation of beliefs can be given. If no consistent explanation can be given then the LNC obviously fails with respect to belief. If we are looking for consistent explanations of other parts of the world then we should look for it within our explanation of beliefs as well.

             It is difficult to understand the claim “The LNC holds (there are no contradictions) and people can and do hold contradictory beliefs.” What is claimed in such a proposition is that there are no contradictions and that there are contradictory beliefs. There is nothing in the world that is a contradiction and yet there are contradictory beliefs. Such a claim is itself a contradiction.

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[1] Girle 2003 also recognizes this, “One major problem with belief logics based on classical logic is that they cannot cope with inconsistent belief, other than by simply rejecting it” (p. 137).

My Response to Sorensen’s proof

Sorensen’s Proof (Believing that (it’s possible to believe that some proposition is impossible and someone believes it) is a true  belief when anyone believes it) 

1. B_a[à($p)($x)(~àp & B_xp)]               (Sorensen’s Premise)

2. ~à($p)($x)(~àp & B_xp)                   (Assume for the sake of a reductio ad absurdum.)

3. ÿ~($p)($x)(~àp & B_xp)                  (2, Rule of ÿ-à Interchange)

4. ÿÿ~($p)($x)(~àp & B_xp)                (3, Axiom S4 ÿp É ÿÿp.)

5. ~àà($p)($x)(~àp & B_xp)                 (2, Rule of ÿ-à Interchange x2.)

6. B_a[à($p)($x)(~àp & B_xp)] &           (1, 5 Conjunction)

~àà($p)($x)(~àp & B_xp) 

Let m name à($p)($x)(~àp & B_xp) 

7. ~àm & B_am                                     (6, synonymous redescription of the proposition with a name.)

8. ($x)(~àm & B_xm)                            (7, Existential Generalization)

9. ($p)($x)(~àp & B_xp)                       (8, Existential Generalization)

10. à($p)($x)(~àp & B_xp)                   (9, Whatever is actual is possible.)

11. ~~à($p)($x)(~àp & B_xp)               (10, By reductio ad absurdum from 2 through 10.) 
 

My response to Sorensen’s Proof: His premise is a contradiction so the argument isn’t valid. 

(C.b*)                                                  If “B_ap” Î m, and if m belongs to a model system W, then there is in W at least one alternative m* to m (with respect to a) such that p Î m*.” (Hintikka, Knowledge and Belief 1962 p. 45)

 

1. B_a[à($p)($x)(~àp & B_xp)]               (Sorensen’s Premise)

2. à($p)($x)(~àp & B_xp) Î m              (1, (C.b*))

3. ($p)($x)(~àp & B_xp) Î m*              (2, since ($p)($x)(~àp & B_xp) is possible in m

                                                            then there must be at least one model (world) such

                                                            that m has access to that model and ($p)($x)(~àp & B_xp)

                                                            is a member of that model)

4. ($x)(~àq & B_xq) Î m*                     (3, Existential Instantiation)

5. ~àq & B_bq Î m*                              (4, Existential Instantiation)

6. ÿ~q Î m*                                         (5, Simplification and Rule of ÿ-à Interchange)

7. B_bq Î m*                                         (5, Simplification)

8. q Î m**                                           (7, (C.b*))

9. ~q Î m**                                           (6, since ~q is necessary in m* then all

                                                                 models (worlds) are such that if m* has 

                                                                access to that model then ~q is a member of that model)

Can I believe p and not p? : Epistemology, Psychology, and Contradictory Beliefs.

(UNL Student Colloquia Submission)

            There is a debate within Philosophy of Mind and Epistemology concerning the notion of Contradictory Beliefs. Several well respected philosophers going as far back as Aristotle to more recent examples of Donald Davidson and Ruth Barcan Marcus have rejected the notion that people can and do have contradictory beliefs. Such a rejection seems to fly in the face of common-sense, so often we seem to find ourselves around people who appear to have contradictory beliefs. What people like Marcus and Davidson appear to recognize is that any attempt to give a theoretical model of belief in an epistemic context (that also holds onto the Law of Non-contradiction) is already doomed to fail if we accept that people can and do have contradictory beliefs. The responses from those who wish to hold onto common-sense have been to either give up on a theoretical model of belief in an epistemic context (Roy Sorensen) or the LNC (Graham Priest).

            I propose another option: there appears to be two different notions of belief that have been conflated so as to produce this debate. The first notion defines belief in terms of assent, a believes that p iff a assents to p. The second notion defines belief in relation to knowledge; beliefs are the kinds of things that epistemic agents gain so they can eventually reach knowledge. Recognizing this distinction will allow us to hold onto our common-sense, that we can have a theoretical model of belief in an epistemic context, and the LNC.

(I really need to thank John Symons for his suggestion that I focus on the Epistemology/Psychology distinction and not just Contradictory Beliefs. It appears to have radically changed the focus of the paper.)

A small difference between the psychological agent and the epistemic agent

Since I agree that there is a difference between the psychological agent and the epistemic agent, I have been trying to figure out where they are different. It seems like one small but important difference between the two agents has to do with what the epistemic agent is focused on. In order for an epsitemic agent to be an epistemic agent she has to be focused or geared towards knowledge. The psychological agent doesn’t necessarily need to be focused on knowledge in order to be a psychological agent. I am sure there are a necessary set of conditions for something to be a psychological agent, but a focus on knowledge is not one of those necessary conditions. To consider a possible example there appear to be instances of psychological agents wanting to give the illusion of having knowledge (say to get a job or impress one’s colleagues), this activity does not appear to be in line with an epistemic agent. At this moment I will not try and claim that the two types of agents are completely distinct, there might be a necessary overlap between the two. Even though it isn’t necessary every time you have a psychological agent you also have an epistemic agent, it might be the case that if you have an epistemic agent then you also have a psychological agent (kind of like the psychological agent has to come first).

Which version of Contradictory Beliefs do we want to reject?

(really sorry again for anyone reading this, I am still not done with this paper) 

I was reading through Armstrong’s discussion of contradictory beliefs in Beliefs, Truth, and Knowledge (1973). I find it interesting that he compares the conjunction

(1) Bap and ~Bap

with the other conjunction

(2) Bap and Ba~p. 

He claims that (1) is an impossible state of affairs while (2) is a possible state of affairs (I think the common-sense version would agree with this). After reading his section, I noticed that his reason for accepting (2) as possible is the notion that people can fail to “…bring [the contradictory beliefs] together, is not aware of holding incompatible beliefs” (p. 105). I realized that within my arguments for denying contradictory beliefs I need to focus on

(3) Ba(p and ~p).

I thought it was interesting that not long after Armstrong starts to discuss (3), he also starts to refer to beliefs as psychological states and also claims “It is a plausible psychological hypothesis that there is more than one belief-system in the human mind, systems which exist in relative isolation from each other” (p. 106). The problem I have with this has to do with when looking at (3) we are looking at one agent, a and whether that one agent can have contradictory beliefs. While we can agree with Armstrong that multiple psychological agents can be present in the human mind (Dissociative Identity Disorder comes to mind, no pun intended) the problem seems to be can one epistemic agent have contradictory beliefs? While we can have Bap and Bb~p where there are two different psychological agents even with the same human mind the questions still need to be asked (1) can we have more than one epistemic agent per person (and if so, what is our justification)? and (2) can an individual epistemic agent have “direct” contradictory beliefs (beliefs of the form (p and ~p))?

One more thing, since I deny contradictory beliefs, I think I should focus on (3) and reserve judgement for (2). I think the problem with (2) is similar to what Armstrong says in that first quote I took. If I reject (2) then it would seem that I have to bring in some notion of logical omniscience (that we automatically have access to all the logical consequences of our beliefs) and even though I have never seen an argument for denying that we logical omniscience, I don’t have an argument for accepting it either. A better way of putting it seems to be just because someone rejects (3) that does not mean that the person has to reject (2).

Psychology and Epistemology, which talks about what?

It was suggested to me by John Symons that there is a difference between psychological beliefs and epistemological beliefs. When considering my concerns about affirming that people can and do have contradictory beliefs there is something very appealling to the suggestion. If we want to hold onto the intuition that people can and do have contradictory beliefs but still recognize that there is something seriously wrong with creating a model that can handle contradictory beliefs in an epistemic context then it seems like such separation would be a possible way to account for this.

Questions continue to pop into my head. When psychology refers to belief and epistemology refers about belief are they talking about the same thing, “belief” just different aspects of it? or are these two distinct things?

I have been trying to think of the issue and realized there is very limited work done in the area. I have been able to come across an interesting paper by Alvin Goldman (“Relation between Epistemology and Psychology” Synthese 1985, Vol 64, p. 29-68) where he appears to give an interesting analysis of the topic. For the moment I am looking at his criticism of logic providing formal rules for epistemology. Goldman says “…so-called ‘rules of inference’ in axiomatic systems or natural deductive systems say nothing about beliefs, or other psychological states” (Goldman p. 42). I think it is important to notice that he has already defined beliefs as a psychological state. If we treat belief in a manner that has little or nothing to do with psychological states, which seems to be in line with someone like Timothy Williamson, “…the point of belief is knowledge” (Knowledge and its Limits p. 1), it would seem that if Williamson is right, beliefs are the kinds of things that point to knowledge then that doesn’t seem very relevant to psychology. The reason I say this is psychology (as far as I know) equates beliefs with behavior (John seemed to think this also). If beliefs are something other than behavior (which it seems like they are) then we cannot say that beliefs (in epistemic contexts) are a psychological state.